If a constant number of moles of a gas is at a pressure of 2.71 atm and has a volume of 85.3 L at a temperature of 173.4 K then what is the final temperature of the gas if the pressure changes to 1.04 atm in a 105.3 L container?
To solve this problem, we can use the ideal gas law equation:
PV = nRT
where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant (0.0821 L.atm/mol.K)
T = temperature in Kelvin
First, we can calculate the initial temperature of the gas using the initial conditions:
P1 = 2.71 atm
V1 = 85.3 L
T1 = 173.4 K
n = constant
Using the ideal gas law equation, we can calculate the initial temperature (T1):
T1 = (P1 * V1) / (n * R)
T1 = (2.71 atm * 85.3 L) / (n * 0.0821 L.atm/mol.K)
Next, we can calculate the final temperature of the gas using the final conditions:
P2 = 1.04 atm
V2 = 105.3 L
Using the same ideal gas law equation with the final conditions, we can calculate the final temperature (T2):
T2 = (P2 * V2) / (n * R)
T2 = (1.04 atm * 105.3 L) / (n * 0.0821 L.atm/mol.K)
Now, we can set the initial and final temperatures equal to each other and solve for n:
(P1 * V1) / (T1 * R) = (P2 * V2) / (T2 * R)
Substitute the known values:
(2.71 atm * 85.3 L) / (T1 * 0.0821 L.atm/mol.K) = (1.04 atm * 105.3 L) / (T2 * 0.0821 L.atm/mol.K)
Solve for T2:
T2 = (1.04 atm * 105.3 L * T1 * 0.0821 L.atm/mol.K) / (2.71 atm * 85.3 L)
Plug in the values for T1 and solve for T2:
T2 = (1.04 atm * 105.3 L * 173.4 K * 0.0821 L.atm/mol.K) / (2.71 atm * 85.3 L)
T2 ≈ 293 K
Therefore, the final temperature of the gas when the pressure changes to 1.04 atm in a 105.3 L container is approximately 293 K.